Evolutionary Algorithms Based on the Automata Theory for ...€¦ · Evolutionary Algorithms Based on the Automata Theory for the Multi-Objective Optimization of Combinatorial Problems

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Evolutionary Algorithms Based on the AutomataTheory for the Multi-Objective Optimization of

Combinatorial Problems

Elias D. NiñoUniversidad del Norte

Colombia

1. Introduction

In this chapter we are going to study metaheuristics based on the Automata Theory for theMulti-objective Optimization of Combinatorial Problems. As well known, CombinatorialOptimization is a branch of optimization. Its domain is optimization problems where theset of feasible solutions is discrete or can be reduced to a discrete one, and the goal is to findthe best possible solution(Yong-Fa & Ming-Yang, 2004). In this field it is possible to find a lotof problems denominated NP-Hard, that is mean that the problem does not have a solutionin Polynomial Time. For instance, problems such as Multi-depot vehicle routing problem(Lim& Wang, 2005), delivery and pickup vehicle routing problem with time windows(Wang& Lang, 2008), multi-depot vehicle routing problem with weight-related costs(Fung et al.,2009), Railway Traveling Salesman Problem(Hu & Raidl, 2008), Heterogeneous, MultipleDepot, Multiple Traveling Salesman Problem(Oberlin et al., 2009) and Traveling Salesmanwith Multi-agent(Wang & Xu, 2009) are categorized as NP-Hard problems.

One of the most classical problems in the Combinatorial Optimization Field is the TravelingSalesman Problem (TSP), it has been analyzed for years(Sauer & Coelho, 2008) either in aMono or Multi-objective way. It is defined as follows: “Given a set of cities and a departurecity, visit each city only once and go back to the departure city with the minimum cost”.Basically, that is mean, visiting each city once, to find an optimal tour in a set of cities, aninstance of TSP problem can be seen in figure 1. Formally, TSP is defined as follows:

minn

∑i=1

n

∑j=1

Cij · Xij (1)

Subject to:n

∑j=1

Xij = 1, ∀i = 1, . . . , n (2)

n

∑j=1

Xij = 1, ∀j = 1, . . . , n (3)

∑i∈κ

∑j∈κ

Xij ≤ |κ| − 1, ∀κ ⊂ {1, . . . , n} (4)

Xij = 0, 1∀i, j (5)

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2 Will-be-set-by-IN-TECH

Where Cij is the cost of the path Xij and κ is any nonempty proper subset of the cities 1, . . . , m.(1) is the objective function. The goal is the optimization of the overall cost of the tour. (2),(3) and (5) fulfills the constrain of visiting each city only once. Lastly, Equation (4) set thesubsets of solutions, avoiding cycles in the tour.

Fig. 1. TSP instance of ten cities

TSP has an important impact on different sciences and fields, for instance in OperationsResearch and Theoretical Computer Science. Most problems related to those fields, are basedin the TSP definition. For instance, problems such as Heterogeneous Machine Scheduling(Kim& Lee, 1998), Hybrid Scheduling and Dual Queue Scheduling(Shah et al., 2009), ProjectManagement(de Pablo, 2009), Scheduling for Multichannel EPONs(McGarry et al., 2008),Single Machine Scheduling(Chunyue et al., 2009), Distributed Scheduling Systems(Yuet al., 1999), Relaxing Scheduling Loop Constraints(Kim & Lipasti, 2003), DistributedParallel Scheduling(Liu et al., 2003), Scheduling for Grids(Huang et al., 2010), ParallelScheduling for Dependent Task Graphs(Mingsheng et al., 2003), Dynamic Scheduling onMultiprocessor Architectures(Hamidzadeh & Atif, 1996), Advanced Planning and SchedulingSystem(Chua et al., 2006), Tasks and Messages in Distributed Real-Time Systems(Manimaranet al., 1997), Production Scheduling(You-xin et al., 2009), Cellular Network for Qualityof Service Assurance(Wu & Negi, 2003), Net Based Scheduling(Wei et al., 2007), SpringScheduling Co-processor(Niehaus et al., 1993), Multiple-resource Periodic Scheduling(Zhuet al., 2003), Real-Time Query Scheduling for Wireless Sensor Networks(Chipara et al., 2007),Multimedia Computing and Real-time Constraints(Chen et al., 2003), Pattern Driven DynamicScheduling(Yingzi et al., 2009), Security-assured Grid Job Scheduling(Song et al., 2006), CostReduction and Customer Satisfaction(Grobler & Engelbrecht, 2007), MPEG-2 TS Multiplexersin CATV Networks(Jianghong et al., 2000), Contention Awareness(Shanmugapriya et al.,2009) and The Hard Scheduling Optimization(Niño, Ardila, Perez & Donoso, 2010) had beenderived from TSP. Although several algorithms have been implemented to solve TSP, there isno one that optimal solves it. For this reason, this chapter discuss novel metaheuristics basedon the Automata Theory to solve the Multi-objective Traveling Salesman Problem.

This chapter is structured as follows: Section 2 shows important definitions to understand theMulti-objective Combinatorial Optimization and the Metaheuristic Approximation. Section3, 4 and 5 discuss Evolutionary Metaheuritics based on the Automata Theory for theMulti-objective Optimization of Combinatorial Problems. Finally, Section 6 and 7 discussthe Experimental Results of each proposed Algorithm using Multi-objective Metrics from thespecialized literature.

82 Real-World Applications of Genetic Algorithms

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Evolutionary Algorithms Based on the Automata Theory for the Multi-Objective Optimization of Combinatorial Problems 3

2. Preliminaries

2.1 Multi-objective optimization

The Multi-objective optimization consists in two or more objectives functions to optimize anda set of constraints. Mathematically, the Multi-objective Optimization model is defined asfollows:

optimize F(X) = { f1(X), f2(X), . . . , fn(X)} (6)

Subject to:H(X) = 0 (7)

G(X) ≤ 0 (8)

Xl ≤ X ≤ Xu (9)

Where F(X) is the set of objective functions, H(X) and G(X) are the constraints of theproblem. Lastly, Xl and Xu are the bounds for the set of variables X.

Unlike to Mono-objective Optimization, Multi-objective Optimization deal with searching aset of Optimal Solutions instead of a Optimal Solution. For instance, table 1 shows threesolutions for a particualr Mono-objective Problem. If we suppose that those are relatedto a maximization problem then the Optimal Solution (found) is the solution 1 otherwise(minimization) will be the solution 2. On the other hand, in table 2 can be seen three solutionsfor a particular Tri-objective Problem. Thus, if we suppose that all the components of thesolutions are related with a minimization problem, solution 2 is a dominated solution due toall the components (0.8, 0.9 and 1.0) are the biggest values. On the other hand, solution0 and 1 are no-dominated solutions due to in the first and second component (0.6 and 0.4)solution 0 is bigger than the relative components of the solution 1 but in the third component(0.5) solution 0 is lower than the same component in solution 1. Both examples show the

k F(Xk)0 101 202 5

Table 1. Solutions for a particular Mono-objective Problem

difference between Mono-objective and Multi-objective Optimization. While the first dealwith finding the Optimal Solution, the last does with finding a set of Optimal Solutions. InCombinatorial Optimization, the set of Optimal Solution is called Pareto Front. It contains allthe no-dominated solutions for a Multi-objective Problem. Figure 2 shows a Pareto Front fora particular Tri-objective Problem. Lastly, it is probably that some Multi-objective Problems

k F(Xk) = { f0(Xk), f1(Xk), f2(Xk)}0 {0.6, 0.4, 0.5}1 {0.2, 0.3, 0.8}2 {0.8, 0.9, 1.0}

Table 2. Solutions for a particular Tri-objective Problem

have an infinite Pareto Front, in those cases is necessary to determinate how many solutionsare required, for instance, using a maximum number of solution permitted in the Pareto Front.

83Evolutionary Algorithms Based on the AutomataTheory for the Multi-Objective Optimization of Combinatorial Problems

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0.60.7

0.80.9

11.1

1.21.3

1.41.5

x 105

6

7

8

9

10

11

12

13

x 104

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x 105

f1f2

f3

Nondominated Solutions

Fig. 2. Pareto Front for a particular Tri-objective Problem

2.2 Tabu search

Tabu Search(Glover & Laguna, 1997) is a basic local search strategy for the Optimization ofCombinatorial Problems. It is defined as follows: Given S as the Initial Solutions Set.

Step 1. Selection. Select x ∈ S

Step 2. Perturbation. Perturbs the solution x for the purpose of knowing its Neighborhood(N(x)). Perturbing a solution means to modify the solution x in order to obtain a new solution

(xi′). The solutions found are called Neighbors, and those represent the Neighborhood.

For instance, figure 3 shows three perturbations for a x solutions and the new solutions

x1′,x2

′and x3

′found. The perturbation can be done according to the representation of the

solutions. Regularly, the representations of the solutions in Combinatorial Problems are basedon Discrete Structures such as Vectors, Matrices, Queues and Lists. Lastly, good solutions areadded to S.

Setp 3. Check Stop Condition. The stop condition can be delimited using rules such as numberof execution without improvement or maximum number of iteration exceeded.

Recently, novels Tabu Search inspired Algorithms have been developed in order tosolve Combinatorial Problems such as Permutation Flow Shop Scheduling(Ren et al.,2011), Displacement based on Support Vector(Fei et al., 2011), Examination TimetablingProblem(Malik et al., 2011), Partial Transmit Sequences for PAPR Reduction(Taspinar et al.,2011), Inverse Problems(An et al., 2011), Fuzzy PD Controllers(Talbi & Belarbi, 2011b),Instrusion Detection(Jian-guang et al., 2011), Tel-Home Care Problems(Lee et al., 2011),Ant Colony inspired Problems(Zhang-liang & Yue-guang, 2011), Steelmaking-ContinuousCasting Production Scheduling(Zhao et al., 2011), Fuzzy Inference System(Talbi & Belarbi,2011a) and Coordination of Dispatchable Distributed Generation and Voltage ControlDevices(Ausavanop & Chaitusaney, 2011).


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Fig. 3. The Neighborhood of a solution x is known after being perturbed

2.3 Genetic algorithms

Genetic Algorithms are Algorithms based on the Theory of Natural Selection(Wijkman, 1996).Thus, Genetic Algorithms mimics the realBehavior Genetic Algorithms(Fisher, 1930) throughthree basic steps: Given a set of Initial Solutions S

Step 1. Selection. Select solutions from a population. In pairs, select two solutions x, y ∈ S

Step 2. Crossover. Cross the selected solutions avoiding local optimums.

Step 3. Mutation. Perturbs the new solutions found for increasing the population. Theperturbation can be done according to the representation of the solution. In this step, goodsolutions are added to S

Figure 4 shows the basics steps of a Genetic Algorithm. The most known Genetic

Fig. 4. Basics steps of a Genetic Algorithm

Algorithms from the literature(Dukkipati & Narasimha Murty, 2002) are the Non-DominatedSorting Genetic Algorithm(Deb et al., 2002) (NSGA-II) and the Strength Pareto EvolutionaryAlgorithm 2(Zitzler et al., 2001; 2002) (SPEA 2). NSGA-II uses a no-dominated sort forsorting the solutions in different Pareto Sets. Consequently, it demands a lot of time, butit allows a global verification of the solutions for avoiding the Local Optimums. On theother hand, SPEA 2 is an improvement of SPEA. The difference with the first version is thatSPEA 2 works using strength for every solution according to the number of solutions that itdominates. Consequently, at the end of the iterations, SPEA 2 has the non dominated solutionsstronger avoiding Local Optimums. SPEA 2 and NSGA-II have been implemented to solvea lot of problems in the Multiobjective and Combinatorial Optimization fiel. For instance,problems such as Pattern-recognition based Machine Translation System(Sofianopoulos &Tambouratzis, 2011), Tuning of Fuzzy Logic controllers for a heating(Gacto et al., 2011),Real-coded Quantum Clones(Xiawen & Yu, 2011), Optimization Problems with CorrelatedObjectives(Ishibuchi et al., 2011), Production Planning(Yu et al., 2011), Optical and DynamicNetworks Designs(Araujo et al., 2011; Wismans et al., 2011), Benchmark multi-objective


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optimization(McClymont & Keedwell, 2011) and Vendor-managed Inventory(Azuma et al.,2011) have been solved using SPEA and NSGA-II.

2.4 Simulated Annealing algorithms

Simulated Annealing(Kirkpatrick et al., 1983) is a generic probabilistic metaheuristic basedin the Annealing in Metallurgy. Similar to Tabu Search, Simulated Annealing explores theneighborhood of solutions being flexible with no-good solutions. That is mean, accepting badsolutions as well as good solution, but only in the first iterations. The acceptation of a badsolution is based on the Boltzmann Probabilistic Distribution:

P(x) = e

(

−(

ETi

))

(10)

Where E is the change of the Energy and Ti is the temperature in the moment i. In the firstlevel of the temperature, bad solutions are accepted as well, anyways, when the temperaturego down, Simulated Annealing behaves similar to Tabu Search (only accept good solutions).

Recentrly, similar to Genetic Algoritms and Tabu Search, many problems have been solvedusing Simulated Annealing metaheuristic. For instance, Neuro Fuzzy - SystemsCzabaski(2006), Contrast Functions for BSSGarriz et al. (2005), Cryptanalysis of TranspositionCipherSong et al. (2008), Transmitter-Receiver Collaborative-Relay BeamformingZheng et al.(2011) and Two-Dimensional Strip Packing ProblemDereli & Sena Da (2007) have been solvedthrough Simulated Annealing inspired algorithms.

2.5 Deterministic Finite Automata

Formally, a Deterministic Finite Automata is a Quint-tuple defined as follows:

A = (Q, Σ, δ, q0, F) (11)

Set of transitions δ. The set of transitions (δ) describes the behavior of the automata. Let a ∈ Sand q, r ∈ Q, then the function is defined as follows:

δ(q, a) = r (12)

Example 1. Let A = (Q, Σ, δ, q0, F), where Q = {q0, q1, q2}, S = {0, 1}, F = {q1} and the set oftransitions δ defined in table 3, the representation of A using a state diagram can be derivedas shown in figure 5. Notice that each state of DFA has transitions with all the elements of Σ.

0 1q0 q2 q0

q1 q1 q1

q2 q2 q1

Table 3. Set of transitions for the DFA of example 1

2.6 Metaheuristic Of Deterministic Swapping (MODS)

Metaheuristic Of Deterministic Swapping (MODS) (Niño et al., 2011) is a local searchstrategy that explores the Feasible Solution Space of a Combinatorial Problem supportedin a data structure named Multi Objective Deterministic Finite Automata (MDFA) (Niño,Ardila, Donoso & Jabba, 2010). A MDFA is a Deterministic Finite Automata that allows the


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Fig. 5. Automata state diagram for the example 1.

representation of the feasible solution space of a Combinatorial Problem. Formally, a MDFAis defined as follows:

M = (Q, Σ, δ, Q0, F(X)) (13)

Where Q represents all the set of states of the automata (feasible solution space), Σ is theinput alphabet that is used for δ (transition function) to explore the feasible solution space ofa combinatorial problem, Q0 contains the initial set of states (initial solutions) and F(X) arethe objectives to optimize.

Example 1. MDFA for a Scheduling Parallel Machine Problem:

A Company has three machines. It is necessary to schedule three processes in parallel P1,P2

and P3. Each process has a duration of 5, 10 y 50 minutes respectively. If the processes canbe executed in any of the machines, how many manners the machines can be assigned to theprocesses? Given the Bi-objective function in (10), what is the optimal Pareto Front?

F(X) =

{

f1(X) =3

∑i=1

i · Xi, f2(X) =3

∑i=1

(

1

i

)

· Xi

}

(14)

First of all, we need to build the MDFA. For doing this, we must define the states of the MDFAsetting the structure of the solution for each state. Therefore, if we state that Xq = (Pk, Pi, Pj)represents the solution for the state q: machine 1 executes the process Pk, machine 2 executesthe process Pi and machine 3 executes the process Pj then the arrays solution for each state

will be Xq0 = (P1, P2, P3), Xq1 = (P1, P3, P2), Xq2 = (P2, P1, P3), Xq3 = (P2, P3, P1), Xq4 =(P3, P1, P2) y Xq5 = (P3, P2, P1). Now, we have six states q0,q1,q2,q3,q4 and q5, those representthe feasible solution space of the Scheduling problem proposed. The set of states for the MDFAof this problem can be seen in figure 6. Once the set of states is defined, the Input Alphabet

Fig. 6. Set of states for the MDFA of example 2

(Σ) and the Transition Function (δ) be done. It is very important to take into account, first,the bond of both allows to perturb the solutions in all the possible manners, in other words,we can change of state using the combination of Σ and δ. Obviously, doing this, we avoid


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unfeasible solutions. Regarding the proposed problem, we propose the set Σ as follows:

Σ = {(P1, P2), (P1, P3), (P2, P3)} (15)

Hence, it is elemental that δ(q0, (P1, P2)) = q2, δ(q0, (P1, P3)) = q5, ... , δ(q5, (P2, P3)) = q3. Atthis part, the transitions has been defined therefore the MDFA can be seen in figure 7.

Finally, the solution of each state is replaced in (10). The results can be seen in table 4 and theOptimal Pareto Front is shown in figure 8.

State Assignments Times F(X)qi M1 M2 M3 M1 M2 M3 f1(X) f2(X)q0 P1 P2 P3 10 50 5 125 36.66q1 P1 P3 P2 10 5 50 170 29.16q2 P2 P1 P3 50 10 5 85 56.66q3 P2 P3 P1 50 5 10 90 55.83q4 P3 P1 P2 5 10 50 175 26.66q5 P3 P2 P1 5 50 10 135 33.33

Table 4. Values of F(X) for the states of example 2

Fig. 7. MDFA for example 2, Parallel execution of processes

As can be seen in figure 7, the feasible solution space for this problem was described using aMDFA. Also, unfeasible solutions are not allowed because of the definition of Σ. Nevertheless,the general problem was not solved, only a particular case of three variables (machines) wasdone. For this reason, it was easy to draw the entire MDFA. However, problems like thisare intractable for a large number of variables, in other words, when the number of variablesgrow the feasible solution space grows exponentially. In this manner, it is not a good ideato draw the entire feasible solution space and pick the best solutions. Thus, what should wedo in order to solve any combinatorial problem, without taking into account its size, using aMDFA? Looking an answer to this question, MODS was proposed.

MODS explores the feasible solution space represented through a MDFA using a searchdirection given by an elitist set of solutions (Q∗). The elitist solution are states that, when


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80 90 100 110 120 130 140 150 160 170 18025

30

35

40

45

50

55

60

f1

f2

No Dominated Solutions

Fig. 8. Pareto Front for the MDFA of example 2, Parallel execution of processes

were visited, their solution dominated at least one solution of an element in Qφ. Qφ containsall the states with non-dominated solutions. Due to this, it can be inferred that the elementsof Q∗ are contained in Qφ, for this reason is true that:

Qφ = Qφ ∪ Q∗ (16)

Lastly, the template algorithm of MODS is defined as follows:

Step 1. Create the initial set of solutions Q0 using a heuristic relative to the problem to solve.

Step 2. Set Qφ as Q0 and Q∗ as φ.

Step 3. Select a random state q ∈ Qφ or q ∈ Q∗

Step 4. Explore the neighborhood of q using δ and Σ. Add to Qφ the solutions found thatare not dominated by elements of Q f . In addition, add to Q∗ those solutions found thatdominated at least one element from Qφ.

Step 5. Check stop condition, go to 3.

3. Simulated Annealing Metaheuristic Of Deterministic Swapping (SAMODS)

Simulated Annealing & Metaheuristic Of Deterministic Swapping(Niño, 2012) (SAMODS) is ahybrid local search strategy based on the MODS theory and Simulated Annealing Algorithmfor the Multiobjective Optimization of combinatorial problems. Its main propose consists inoptimizing a combinatorial problem using a Search Direction and an Angle Improvement.SAMODS is based in the next Automata:

M = (Q, Q0, P(q), F(X), A(n)) (17)

Alike MODS, Q0 is the set of initial solutions, Q is the feasible solution space and F(X) are thefunctions of the combinatorial problem. P(q) and A(n) are defined as follows:

P(q) is the Permutation Function, formally it is defined as follows:

P(q) : Q → Q (18)

P receives a solution q ∈ Q and perturbs it returning a new solution ri ∈ Q. The perturbationcan be done based on the representation of the solutions. An example of some perturbationsbased on the representation of the solution can be seen in figure 15.


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Fig. 9. Different representation and perturbation of solutions.

A(n) is the Weight Function. Formally, it is defined as follow:

A(n) : N → ℜn (19)

Where n is the number of objectives of the problem.

Function A receives a natural number as parameter and it returns a vector with the weights.The weight values are randomly generated with an uniform distribution. Those represent theweight to assign to each function of the combinatorial problem. The weight values returnedby the function fulfill the next constrain:

n

∑i=1

αi = 1, 0 ≤ αi ≤ 1 (20)

Where αi is the weight assigned to function i. Table 5 shows some vectors randomly generatedby A(n).

Input Parameter Function Vector of Weights2 A(2) {0.6, 0.4}3 A(3) {0.2, 0.4, 0.4}4 A(4) {0.3, 0.8, 0.1, 0.0}

Table 5. Some weight vectors generated by A(n)

But, what is the importance of those weights? The weights, in an implicit manner, allowsetting the angle direction to the solutions. The angle direction is the course being followedby the solutions for optimizing F(X). Hence, when the weights values are changed, the angle ofoptimization is changed and a new search direction is obtained. For instance, different searchdirections for different weight values are shown in figure 16 in a Bi-objective combinatorialproblem. Due to this, (6) is rewritten as follows:

F(X) =n

∑i=1

αi · fi(X) (21)

Where n is the number of objectives of the problem and αi is the weight assigned to thefunction i. The weights fulfills the constrain established in (20).

SAMODS main idea is simple: it takes advantage of the search directions given by MODSand it proposed an angle direction given by the function A(n). Thus, there are two directions;the first helps in the convergence of the Pareto Front and the second helps the solutions tofind neighborhoods where F(X) is optimized. Due to this, SAMODS template is defined asfollows:

Step 1. Setting sets. Set Q0 as the set of Initial Solutions. Set Qφ and Q∗ as Q0.


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Fig. 10. Different angles given by different weights for a Bi-objective Problem.

Step 2. Settings parameters. Set T as the initial temperature, n as the number of objectives ofthe problem and ρ as the cooler factor.

Step 3. Setting Angle. If T is equal to 0 then got to 8, else set Ti+1 = ρ × Ti, randomly selects ∈ Qφ, set W = A(n) = {w1, w2, · · · , wn} and go to step 4.

Step 4. Perturbing Solutions. Set s′= P(s), add to Qφ and Q∗ according to the next rules:

Qφ = Qφ ∪{

s′}

⇔ ( � ∃r ∈ Qφ)(r dominated to s′) (22)

Q∗ = Q∗ ∪{

s′}

⇔ (∃r ∈ Q∗)(s′

dominated to r) (23)

If Qφ has at least one element that dominated to s′

go to step 5, otherwise go to step 7.

Step 5. Guess with dominated solutions. Randomly generated a number n ∈ [0, 1]. Set z asfollows:

z = e(−(γ/Ti)) (24)

Where Ti is the temperature value in moment i and γ is defined as follows:

γ =n

∑i=1

wi · fi(sX)−n

∑i=1

wi · fi(s′

X) (25)

Where sX is the vector X of solution s, s′

X is the vector X of solution s′, wi is the weight

assigned to the function i and n is the number of objectives of the problem. If n < z then set s

as s′

and go to step 4 else go to step 6.

Step 6. Change the search direction. Randomly select a solution s ∈ Q∗ and go to step 4.

Step 7. Removing dominated solutions. Remove the dominated solution for each set (Q∗ andQφ). Go to step 3.

Step 8. Finishing. Qφ has the non-dominated solutions.

As can be seen in figure 11, alike MODS, SAMODS removes the dominated solutions whenthe new solution found is not dominated. Besides, if the new solution found dominatedat least one element from the solution set (Qφ) then it will be added to the elitisms set(Q∗) that works as a search direction for the Pareto Front. As far as here, SAMODS couldsounds as a simple local search strategy but not, when a new solution found is dominated,SAMODS tries to improve it using guessing. Guessing is done accepting dominated solutionas good solutions. Alike Simulated Annealing inspired algorithms, the dominated solutionsare accepted under the Boltzmann Distribution Probability assigning weights to the objectivesof the problem. It is probably that perturbing a dominated solution, a non-dominated solutioncan be found as can be seen in figure 12. Due to this, local optimums are avoided. When the


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temperature is low, the bad solutions are avoided because z value is low therefore SAMODSaccepts only non-dominated solutions. However, by that time, Qφ will be leaded on by Q∗.

Fig. 11. Behavior of SAMODS when the new solution found is not dominated. Once a newsolution found is non-dominated, it is added to the elitism set Q∗ and the dominatedsolutions from Qφ are removed.

Fig. 12. Behavior of SAMODS when the new solution found is dominated. In this case,guessing gives a new solution non-dominated.

4. Genetic Simulated Annealing Metaheuristic Of Deterministic Swapping

(SAGAMODS)

Simulated Annealing, Genetic Algorithm & Metaheuristic Of Deterministic Swapping(Niño,2012) (SAGAMODS) is a hybrid search strategy based on the Automata Theory, SimulatedAnnealing and Genetics Algorithms. SAGAMODS is an extension of the SAMODS theory.It comes up as result of the next question: could SAMODS avoid quickly local optimums?Although, SAMODS avoids local optimums guessing, it can take a lot of time acceptingdominated solutions for finding non-dominated. Thus, the answer to this question is basedon the Evolutionary Theory. SAGAMODS proposes crossover step before SAMODS templateis executed. Due to this, SAGAMODS supports to SAMODS for exploring distant regions ofthe solution space.

Formally, SAGAMODS is based on the next automata:

M = (Q, QS, C(q, r, k), F(X)) (26)


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Where Q is the feasible solutions space, QS is the initial solutions and F(X) are the objectivesof the problem. C(q, r, k) is defined as follows:

Formally, Cross Function K is defined as follows:

C(q, r, k) : Q → Q (27)

Where q, r ∈ Q and k ∈ N. q and r are named parents solutions and k is the cross point. Themain idea of this function is cross two solutions in the same point and returns a new solution.For instance, two solutions of 4 variables are cross in figure 13. Obviously, the crossover ismade regarding the representation of the solutions. Lastly, SAGAMODS template is defined

Fig. 13. Crossover between two solutions. Solutions of the states qk and qj are crossed inorder to get state qi

as follows:

Step 1. Setting parameters. Set QS as the solution set, x as the number of solutions to cross foreach iteration.

Step 2. Selection. Set QC (crossover set) as selection of x solutions in QS, QM (mutation set) asφ and k as a random value.

Step 3. Crossover. For each si, si+1 ∈ QC/1 ≤ i < |QC|:

QM = QM ∪ {C(si, si+1, k)} (28)

Step 4. Mutation. Set Q0 as QM. Execute SAMODS as a local search strategy.

Step 5. Check stop conditions. Go to 2.

5. Evolutionary Metaheuristic Of Deterministic Swapping (EMODS)

Evolutionary Metaheuristic of Deterministic Swapping (EMODS), is a novel framework thatallows the Multiobjective Optimization of Combinatorial Problems. Its framework is based onMODS template therefore its steps are the same: create Initial Solutions, Improve the Solutions(Optional) and Execute the Core Algorithm. Unlike SAMODS and SAGAMODS, EMODSavoids the slowly convergence of Simulated Annealing’s method. EMODS explores differentregions from the feasible solution space and search for non-dominated solution using TabuSearch.

The Core Algorithm is defined as follows:


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Step 1. Set θ as the maximum number of iterations, β as the maximum number of state selectedin each iteration, ρ as the maximum number of perturbations by state and Qφ as Q0

Step 2. Selection. Randomly select a state q ∈ Qφ or q ∈ Q∗

Step 3. Mutation - Tabu Search. Set N as the new solutions found as result of perturbing q. Addto Qφ and Q∗ according to the next equations:

(

Qφ = Qφ ∪ {q})

⇐⇒(

� ∃r ∈ Qφ/q is dominated by r)

(29)

(Q∗ = Q∗ ∪ {q}) ⇐⇒(

∃r ∈ Qφ/r is dominated by q)

(30)

Remove the states with dominated solutions for each set.

Step 4. Crossover. Randomly select states from Qφ and Q∗. Generate a random point of cross.

Step 5. Check stop condition, go to 3.

Step 2 and 3 support the algorithm in removing dominated solutions from the set of solutionsQφ as can be seen in figure 3. However, one of the most important steps in the EMODSalgorithm is step 4. There, similar to SAGAMODS, the algorithm applies an EvolutionaryStrategy based in the crossover step of Genetic Algorithms for avoiding Local Optimums.Due to the crossover is not always made in the same point (the k-value is randomly generatedin each state analyzed) the variety of solutions found are diverse avoiding local optimums. Anoverview of EMODS behavior for a Tri-objective Combinatorial Optimization problem can beseen in figure 14

6. Experimental analysis

6.1 Experimental settings

The algorithms were tested using well-known instances from the Multi-objective TravelingSalesman Problem taken from TSPLIB(Heidelberg, n.d.). The instances worked are shown intable 6 and the input parameters for the algorithms are shown in table 7. The test of thealgorithms was made using a Dual Core Computer with 2 Gb RAM. The optimal solutionswere constructed based in the best non-dominated solutions of all algorithms in comparisonfor each instance worked.

6.2 Performance metrics

There are metrics that allow measuring the quality of a set of optimal solutions and theperformance of an Algorithm (Corne & Knowles, 2003). Most of them use two Pareto Fronts.The first one is PFtrue and it refers to the real optimal solutions of a combinatorial problem.The second is PFknow and it represents the optimal solutions found by an algorithm.

Generation of Non-dominated Vectors (GNDV) It measures the number of No DominatesSolutions generated by an algorithm.

GNDV = |PFknow| (31)

A higher value for this metric is desired. Rate of Generation of No-dominated Vectors (RGNDV)This metric measures the proportion of the No Dominates Solutions (31) generated by an


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Fig. 14. An overview of EMODS behavior for a Tri-objective Problem.

algorithm and the Real Solutions.

RGNDV =

(

GNDV

|PFtrue|

)

· 100% (32)

A value closer to 100% for this metric is desired. Real Generation of Non-dominated Vectors(ReGNDV) This metric measures the number of Real Solutions found by an algorithm.

ReGNDV = |{y|y ∈ PFknow ∧ y ∈ PFtrue}| (33)


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Combinatorial Problem Instance Number of Objectives

KROAB100

2

KROAC100

KROAD100

KROAE100

KROBC100

KROBD100

KROBE100

KROCD100

KROCE100

KRODE100

KROABC100

3

KROABD100

KROABE100

KROACD100

KROACE100

KROADE100

KROBCD100

KROBCE100

KROBDE100

KROCDE100

KROABCD100

4

KROABCE100

KROABDE100

KROACDE100

KROBCDE100

KROABCDE100 5

Table 6. Instances worked for testing the proposed algorithms.

Algorithm Max. Iterations Max. Perturbations Initial Temperature Cooler Value Crossover Rate

MODS 100 80 NA NA NA

SAMODS 100 80 1000 0.95 NA

SAGAMODS 100 80 1000 0.95 0.6

EMODS 100 80 NA NA 0.6

Table 7. Parameters setting for each compared algorithm.

A value closer to |PFtrue| for this metric is desired.

Generational Distance (GD) This metric measures the distance between PFknow and PFtrue. Itallows to determinate the error rate in terms of the distance of a set of solutions relative to thereal solutions.

GD =

(

1

|PFknow|

)

·

(

|PFknow|

∑i=1

di

)(1/p)

(34)


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Where di is the smallest Euclidean distance between the solution i of FPknow and the solutionsof FPtrue. p is the dimension of the combinatorial problem, it means the number of objectivefunctions. Inverse Generational Distance (IGD) This is another distance measurement betweenFPknow and FPtrue:

IGD =

(

1

|PFtrue|

)

·

(

|PFknow|

∑i=1

di

)

(35)

Where di is the smallest Euclidean distance between the solution i of PFknow and the solutionsof PFtrue. Spacing (S) It measures the range variance of neighboring solutions in PFknow

S =

(

1

|PFknow| − 1

)2

·

(

|PFknow|

∑i=1

(

d − di

)2)(1/p)

(36)

Where di is the smallest Euclidean distance between the solution i of PFknow and the rest ofsolutions of PFknow. d is the mean of all di. p is the dimension of the combinatorial problem.

A value closer to 0 for this metric is desired. A value of 0 means that all the solutions areequidistant.

Error Rate (ε) It estimates the error rate respect to the precision of the Real Algorithms Solutions(33) as follows:

ε =

(∣

∣

∣

∣

PFtrue

ReGNDV

∣

∣

∣

∣

)

· 100% (37)

A value of 0% in this metric means that the values of the Real Pareto Front are constructedfrom the values of the Algorithm Pareto Front.

Lastly, notice that every metric by itself does not have sense. It is necessary to support inthe other metrics for a real judge about the quality of the solutions. For instance, if a ParetoFront has a higher value in GNDV but a lower value in ReGNDV then the solutions has apoor-quality.

6.3 Experimental results

The tests made with Bi-objectives, Tri-objectives, Quad-objectives and Quin-objectives TSPinstances are shown in tables 8, 9, 10 and 11 respectively. The average of the measurementis shown in table 12. Furthermore, a graphical comparison for bi-objectives and tri objectivesinstances worked is shown in figures 15 and 16 respectively.

6.4 Analysis

It can be concluded, that, in the case of two and three objectives, metrics such as S, IGD,GD and ε determine the best algorithm. In this case, the measurement of the metrics issimilar for SAMODS and SAGAMODS. On the other hand, MODS has the most poor-qualitymeasurement for the metrics used and EMODS has the best quality measurement for the samemetrics.

Lastly, why are the results of the metrics similar for quint-instances? In this case, all thesolutions for each solution set are in the optimal set. The answer to this question is based inthe angle improvement. MODS as a local search strategy explore a part of the feasible solutionusing its search direction (Q∗). However, SAMODS and SAGAMODS, in addition, use asearch direction given by the change of the search angle. While SAMODS was looking in a


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Fig. 15. Graphical comparison between MODS, SAMODS, SAGAMODS and EMODS forBi-objective TSP instances.

Fig. 16. Graphical comparison between MODS, SAMODS, SAGAMODS and EMODS forTri-objective TSP instances.

part of the feasible solution space, SAGAMODS was doing the same in other. The same reasonapplies to EMODS. It can be possible because of the large size of the feasible solution space(ℜ5). The possibility of exploring the same part of the solution space for different algorithmsis low.


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Instance Algorithm GNDV RGNDV ReGNDV(

ReGNDVGNDV

)

% S GD IGD ε

AB

MODS 289 0.0189 0 0% 0.0193 21.2731 2473.4576 100%

SAMODS 7787 0.5096 1247 16.01% 0.001 0.2404 229.2593 91.84%

SAGAMODS 8479 0.5549 2974 35.07% 0.0007 0.1837 158.8229 80.54%

EMODS 26125 1.7096 11060 42.33% 0.0002 0.0412 75.8814 27.62%

AC

MODS 217 0.0155 0 0% 0.034 28.575 2751.3232 100%

SAMODS 6885 0.4927 2303 33.45% 0.0008 0.2297 179.0354 83.52%

SAGAMODS 7023 0.5025 2431 34.61% 0.0008 0.2628 243.7536 82.6%

EMODS 20990 1.502 9241 44.03% 0.0002 0.0617 119.8825 33.87%

AD

MODS 281 0.0187 0 0% 0.0139 20.4429 2198.883 100%

SAMODS 6383 0.4253 1464 22.94% 0.0029 0.3188 275.9153 90.24%

SAGAMODS 6289 0.4191 764 12.15% 0.0016 0.2835 211.7935 94.91%

EMODS 17195 1.1458 12779 74.32% 0.0005 0.0521 53.5314 14.85%

AE

MODS 283 0.0189 0 0% 0.0533 21.3238 2433.63 100%

SAMODS 5693 0.3804 1433 25.17% 0.0016 0.4308 402.0483 90.42%

SAGAMODS 6440 0.4304 1515 23.52% 0.0013 0.3906 422.7859 89.88%

EMODS 20695 1.383 12016 58.06% 0.0002 0.066 124.7197 19.7%

BC

MODS 298 0.0212 0 0% 0.0158 19.537 2411.8365 100%

SAMODS 6858 0.488 789 11.5% 0.0024 0.3597 433.0882 94.39%

SAGAMODS 6919 0.4923 2201 31.81% 0.0015 0.2378 192.5601 84.34%

EMODS 21902 1.5584 11064 50.52% 0.0003 0.0582 115.5673 21.28%

BD

MODS 241 0.0198 0 0% 0.0239 21.7441 2251.0972 100%

SAMODS 6844 0.561 2054 30.01% 0.0021 0.2542 248.0971 83.16%

SAGAMODS 5934 0.4864 1971 33.22% 0.0018 0.2818 229.2093 83.84%

EMODS 19420 1.5919 8174 42.09% 0.0003 0.0432 57.6434 32.99%

BE

MODS 280 0.0259 0 0% 0.0309 19.0193 2622.5243 100%

SAMODS 6260 0.5789 952 15.21% 0.001 0.2601 245.1433 91.2%

SAGAMODS 5802 0.5365 1622 27.96% 0.0025 0.3912 476.4848 85%

EMODS 17362 1.6055 8240 47.46% 0.0004 0.0631 111.0209 23.8%

CD

MODS 286 0.022 0 0% 0.0184 18.0035 2040.9722 100%

SAMODS 6171 0.4751 1912 30.98% 0.0007 0.2588 196.3394 85.28%

SAGAMODS 6301 0.4851 994 15.78% 0.0014 0.2852 248.5855 92.35%

EMODS 18628 1.434 10084 54.13% 0.0002 0.0426 48.3785 22.37%

CE

MODS 224 0.0187 0 0% 0.0285 23.1312 2245.6542 100%

SAMODS 5881 0.4919 946 16.09% 0.0017 0.2894 242.2535 92.09%

SAGAMODS 4613 0.3859 939 20.36% 0.0028 0.481 411.854 92.15%

EMODS 15211 1.2724 10070 66.2% 0.0003 0.0339 22.2645 15.77%

DE

MODS 228 0.0147 0 0% 0.0477 23.6222 1864.9602 100%

SAMODS 6110 0.3928 1157 18.94% 0.0022 0.2942 207.7947 92.56%

SAGAMODS 7745 0.4979 407 5.26% 0.0012 0.2644 269.6204 97.38%

EMODS 20058 1.2896 13990 69.75% 0.0005 0.0304 23.8829 10.06%

Table 8. Measuring algorithms performance for Bi-objectives instances of Traveling SalesmanProblem with Multi-objective optimization metrics.


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ReGNDVGNDV

)

% S GD IGD ε

ABC

MODS 2115 0.0307 83 3.92% 0.1567 0.2819 3075.4309 99.88%

SAMODS 12768 0.1853 227 1.78% 0.0722 0.0421 2256.4593 99.67%

SAGAMODS 12523 0.1818 328 2.62% 0.073 0.0427 2220.5614 99.52%

EMODS 70474 1.023 68254 96.85% 0.0477 0.001 5.6388 0.93%

ABD

MODS 1951 0.0292 74 3.79% 0.1524 0.305 3153.9212 99.89%

SAMODS 12094 0.1811 317 2.62% 0.0746 0.0441 2270.3475 99.53%

SAGAMODS 12132 0.1817 250 2.06% 0.0726 0.0441 2286.7374 99.63%

EMODS 68001 1.0184 66133 97.25% 0.0471 0.0011 6.3212 0.96%

ABE

MODS 1931 0.0281 63 3.26% 0.1496 0.315 3278.1554 99.91%

SAMODS 12461 0.1815 373 2.99% 0.0743 0.0438 2371.7641 99.46%

SAGAMODS 12391 0.1805 370 2.99% 0.0745 0.0436 2304.3277 99.46%

EMODS 70411 1.0257 67839 96.35% 0.0474 0.0012 8.0639 1.17%

ACD

MODS 2031 0.0305 66 3.25% 0.1425 0.2945 3213.7378 99.9%

SAMODS 12004 0.1802 241 2.01% 0.0734 0.0444 2277.0343 99.64%

SAGAMODS 12123 0.182 206 1.7% 0.0735 0.0442 2310.3683 99.69%

EMODS 67451 1.0127 66090 97.98% 0.0468 0.001 4.5012 0.77%

ACE

MODS 1950 0.0306 57 2.92% 0.1628 0.3024 3215.8357 99.91%

SAMODS 11382 0.1785 263 2.31% 0.074 0.0461 2271.6542 99.59%

SAGAMODS 11476 0.18 303 2.64% 0.0734 0.0456 2241.9933 99.52%

EMODS 64804 1.0162 63145 97.44% 0.048 0.0012 7.3103 0.98%

ADE

MODS 1824 0.0274 67 3.67% 0.1487 0.3289 3248.597 99.9%

SAMODS 12149 0.1827 179 1.47% 0.0733 0.0442 2336.2798 99.73%

SAGAMODS 11773 0.1771 258 2.19% 0.0771 0.0457 2346.6414 99.61%

EMODS 67767 1.0193 65981 97.36% 0.0468 0.0011 5.7824 0.76%

BCD

MODS 2065 0.03 43 2.08% 0.1451 0.2927 3206.9305 99.94%

SAMODS 13129 0.1908 260 1.98% 0.0712 0.0417 2387.8219 99.62%

SAGAMODS 12889 0.1873 253 1.96% 0.0786 0.042 2308.1811 99.63%

EMODS 70035 1.0176 68270 97.48% 0.0452 0.001 5.6235 0.81%

BCE

MODS 2009 0.0286 58 2.89% 0.1505 0.3065 3327.787 99.92%

SAMODS 12992 0.1852 229 1.76% 0.0701 0.0428 2448.3577 99.67%

SAGAMODS 12582 0.1794 201 1.6% 0.0736 0.0445 2503.0421 99.71%

EMODS 71176 1.0147 69654 97.86% 0.0464 0.0011 7.8122 0.7%

BDE

MODS 2039 0.0316 45 2.21% 0.1532 0.2914 3252.7813 99.93%

SAMODS 12379 0.192 205 1.66% 0.0728 0.0434 2401.8804 99.68%

SAGAMODS 12427 0.1928 195 1.57% 0.0742 0.0431 2377.0621 99.7%

EMODS 65509 1.0163 64015 97.72% 0.0476 0.0011 5.6322 0.69%

CDE

MODS 2010 0.0278 83 4.13% 0.1463 0.3022 3094.2824 99.89%

SAMODS 13084 0.1807 399 3.05% 0.0712 0.0414 2193.6586 99.45%

SAGAMODS 13009 0.1796 347 2.67% 0.0719 0.0418 2224.4738 99.52%

EMODS 74063 1.0227 71589 96.66% 0.0453 0.0011 7.2279 1.14%

Table 9. Measuring algorithms performance for Tri-objective instances of TSP withMulti-objective optimization metrics.


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ReGNDVGNDV

)

% S GD IGD ε

ABCD

MODS 5333 0.0925 3303 61.94% 0.3497 0.0256 6030.5288 94.27%

SAMODS 28523 0.4947 12178 42.7% 0.231 0.0042 3454.1238 78.88%

SAGAMODS 36802 0.6382 14967 40.67% 0.2203 0.0031 3092.5232 74.04%

EMODS 201934 3.502 27214 13.48% 0.1754 0.0005 1991.148 52.8%

ABCE

MODS 5533 0.0973 3439 62.15% 0.3452 0.0244 5861.8605 93.95%

SAMODS 27684 0.4868 11471 41.44% 0.2331 0.0043 3444.6454 79.83%

SAGAMODS 35766 0.6289 14552 40.69% 0.2232 0.0032 3118.6397 74.41%

EMODS 204596 3.5976 27408 13.4% 0.1754 0.0005 1885.4464 51.81%

ABDE

MODS 5259 0.0942 3142 59.75% 0.3487 0.0256 5864.5036 94.37%

SAMODS 27180 0.4869 11247 41.38% 0.232 0.0043 3398.9429 79.85%

SAGAMODS 34930 0.6257 14472 41.43% 0.2236 0.0033 2986.5319 74.08%

EMODS 195756 3.5067 26963 13.77% 0.1775 0.0005 1916.7141 51.7%

ACDE

MODS 5466 0.094 3400 62.2% 0.3405 0.0246 5617.5202 94.15%

SAMODS 26757 0.4602 11336 42.37% 0.235 0.0044 3394.8396 80.5%

SAGAMODS 34492 0.5932 14638 42.44% 0.2265 0.0033 2965.4482 74.83%

EMODS 196800 3.3845 28774 14.62% 0.1764 0.0005 1793.4489 50.52%

BCDE

MODS 5233 0.0879 3082 58.9% 0.3499 0.0259 5677.9988 94.83%

SAMODS 28054 0.471 11739 41.84% 0.2315 0.0042 3296.196 80.29%

SAGAMODS 36258 0.6087 15145 41.77% 0.2218 0.0032 2902.8041 74.57%

EMODS 203017 3.4083 29599 14.58% 0.1752 0.0005 1873.1748 50.31%

Table 10. Measuring algorithms performance for Quad-objectives instances of TSP withMulti-objective optimization metrics.


ReGNDVGNDV

)

% S GD IGD ε

ABCDE

MODS 7517 0.0159 7517 100% 0.5728 0.0125 15705.6864 98.41%

SAMODS 26140 0.0554 26140 100% 0.4101 0.0033 10801.6382 94.46%

SAGAMODS 26611 0.0564 26611 100% 0.4097 0.0033 10544.8901 94.36%

EMODS 411822 0.8723 411822 100% 0.3136 0.0001 950.4252 12.77%

Table 11. Measuring algorithms performance for Quint-objectives instances of TSP withMulti-objective optimization metrics.


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Objectives Algorithm GNDV RGNDV ReGNDV(

ReGNDVGNDV

)

% S GD IGD ε

2

MODS 262.7 0.0194 0 0% 0.0286 21.6672 2329.4338 100%

SAMODS 6487.2 0.4796 1425.7 22.03% 0.0016 0.2936 265.8974 89.47%

SAGAMODS 6554.5 0.4791 1581.8 23.97% 0.0015 0.3062 286.547 88.3%

EMODS 19758.6 1.4492 10671.8 54.89% 0.0003 0.0492 75.2773 22.23%

3

MODS 1992.5 0.0295 63.9 3.21% 0.1508 0.302 3206.7459 99.91%

SAMODS 12444.2 0.1838 269.3 2.16% 0.0727 0.0434 2321.5258 99.6%

SAGAMODS 12332.5 0.1822 271.1 2.2% 0.0743 0.0437 2312.3389 99.6%

EMODS 68969.1 1.0187 67097 97.3% 0.0468 0.0011 6.3914 0.89%

4

MODS 5364.8 0.0932 3273.2 60.99% 0.3468 0.0252 5810.4824 94.31%

SAMODS 27639.6 0.4799 11594.2 41.94% 0.2325 0.0043 3397.7495 79.87%

SAGAMODS 35649.6 0.619 14754.8 41.4% 0.2231 0.0032 3013.1894 74.39%

EMODS 200420.6 3.4798 27991.6 13.97% 0.176 0.0005 1891.9864 51.43%

5

MODS 7517 0.0159 7517 100% 0.5728 0.0125 15705.6864 98.41%

SAMODS 26140 0.0554 26140 100% 0.4101 0.0033 10801.6382 94.46%

SAGAMODS 26611 0.0564 26611 100% 0.4097 0.0033 10544.8901 94.36%

EMODS 411822 0.8723 411822 100% 0.3136 0.0001 950.4252 12.77%

Table 12. Measuring algorithms performance for Multi-objectives instances of TSP withMulti-objective optimization metrics.

7. Conclusion

SAMODS, SAGAMODS and EMODS are algorithms based on the Automata Theory for theMulti-objective Optimization of Combinatorial Problems. All of them are derived from theMODS metaheuristic, which is inspired in the Theory of Deterministic Finite Swapping.SAMODS is a Simulated Annealing inspired Algorithm. It uses a search direction in orderto optimize a set of solution (Pareto Front) through a linear combination of the objectivefunctions. On the other hand, SAGAMODS, in addition to the advantages of SAMODS, isan Evolutionary inspired Algorithm. It implements a crossover step for exploring far regionsof a solution space. Due to this, SAGAMODS tries to avoid local optimums owing to it takesa general look of the solution space. Lastly, in order to avoid slow convergence, EMODS isproposed. Unlike SAMODS and SAGAMODS, EMODS does not explore the neighborhood ofa solution using Simulated Annealing, this step is done using Tabu Search. Thus, EMODS getsoptimal solution faster than SAGAMODS and SAMODS. Lastly, the algorithms were testedusing well known instances from TSPLIB and metrics from the specialized literature. Theresults shows that for instances of two, three and four objectives, the proposed algorithmhas the best performance as the metrics values corroborate. For the last instance worked,

quint-objective, the behavior of MODS, SAMODS and SAGAMODS tend to be the same, themhave similar error rate but, EMODS has a the best performance. In all the cases, EMODS showsthe best performance. However, for the last test, all the algorithms have different solutions setsof non-dominated solutions, and those form the optimal solution set.


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8. Acknowledgment

First of all, I want to thank to God for being with me in my entire life, He made this possible.

Secondly, I want to thank to my parents Elias Niño and Arely Ruiz and my sister CarmenNiño for their enormous love and support. Finally, and not less important, to thank to mybeautiful wife Maria Padron and our baby for being my inspiration.

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Real-World Applications of Genetic AlgorithmsEdited by Dr. Olympia Roeva

ISBN 978-953-51-0146-8Hard cover, 376 pagesPublisher InTechPublished online 07, March, 2012Published in print edition March, 2012

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The book addresses some of the most recent issues, with the theoretical and methodological aspects, ofevolutionary multi-objective optimization problems and the various design challenges using different hybridintelligent approaches. Multi-objective optimization has been available for about two decades, and itsapplication in real-world problems is continuously increasing. Furthermore, many applications function moreeffectively using a hybrid systems approach. The book presents hybrid techniques based on Artificial NeuralNetwork, Fuzzy Sets, Automata Theory, other metaheuristic or classical algorithms, etc. The book examinesvarious examples of algorithms in different real-world application domains as graph growing problem, speechsynthesis, traveling salesman problem, scheduling problems, antenna design, genes design, modeling ofchemical and biochemical processes etc.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

Elias D. Niño (2012). Evolutionary Algorithms Based on the Automata Theory for the Multi-ObjectiveOptimization of Combinatorial Problems, Real-World Applications of Genetic Algorithms, Dr. Olympia Roeva(Ed.), ISBN: 978-953-51-0146-8, InTech, Available from: http://www.intechopen.com/books/real-world-applications-of-genetic-algorithms/evolutionary-algorithms-based-on-the-automata-theory-for-the-multi-objective-optimization-of-combina

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